Analysis of Single Server Repairable Queueing System With Many Kinds of Service, Single Working Vacation and Standby Server
DOI:
https://doi.org/10.2298/YJOR230915029BKeywords:
Markovian arrival process, phase-type distribution, standby, breakdown, single working vacationAbstract
We consider a single server queueing model in which the customers arrive according to a Markovian Arrival Process (MAP). We modeled this queueing system with many kinds of service where the customers can choose any kind from the server. Each kind of service follows Phase-type (PH) distribution. We analyzed the model with breakdown, repair, single working vacation and standby server. A standby server works only when the main server is breakdown while serving the customer during a busy or vacation period. Using the matrix-geometric approach, we examined the model. A few real-world examples have been given, and several graphical and numerical examples have also been provided.References
M. Neuts, Matrix-geometric Solutions in Stochastic Models: An Algorithmic Approach, ser. Algorithmic Approach. Dover Publications, 1994. [Online]. Available: https://books.google.co.in/books?id=WPol7RVptz0C
S. R. Chakravarthy, “Markovian arrival processes,” Wiley encyclopedia of operations research and management science, 2010.
S. R. Chakravarthy, Introduction to Matrix-Analytic Methods in Queues 2: Analytical and Simulation Approach-Queues and Simulation. John Wiley & Sons, 2022.
L. D. Servi and S. G. Finn, “M/M/1 queues with working vacations (M/M/1/WV),” Performance Evaluation, vol. 50, no. 1, pp. 41-52, 2002. doi: https://doi.org/10.1016/S0166-5316(02)00057-3
S. R. Chakravarthy, “A multi-server synchronous vacation model with thresholds and a probabilistic decision rule,” European journal of operational research, vol. 182, no. 1, pp. 305-320, 2007.
S. R. Chakravarthy, “Analysis of a multi-server queue with markovian arrivals and synchronous phase type vacations,” Asia-Pacific Journal of Operational Research, vol. 26, no. 01, pp. 85-113, 2009. doi: https://doi.org/10.1142/S0217595909002134
S. R. Chakravarthy, “Analysis of a multi-server queueing model with vacations and optional secondary services,” Mathematica Applicanda, vol. 41, no. 2, 2013.
B. T. Doshi, “Queueing systems with vacations-a survey,” Queueing systems, vol. 1, pp. 29-66, 1986.
W. M. Kempa, “Transient workload distribution in the M/G/1 finite-buffer queue with single and multiple vacations,” Annals of Operations Research, vol. 239, no. 2, pp. 381-400, 2016. doi: https://doi.org/10.1007/s10479-015-1804-x
N. shuo Tian and Z. G. Zhang, “Vacation queueing models theory and applications,” Operations Research and Management Science, 2006.
“MAP/PH/1 queue with working vacations, vacation interruptions and N policy,” Applied Mathematical Modelling, vol. 37, no. 6, pp. 3879-3893, 2013. doi: https://doi.org/10.1016/j.apm.2012.07.054
C. Goswami and N. Selvaraju, “A working vacation queue with priority customers and vacation interruptions,” International Journal of Operational Research, vol. 17, no. 3, pp. 311-332, 2013. doi: https://doi.org/10.1504/IJOR.2013.054438
V. Chandrasekaran, K. Indhira, M. Saravanarajan, and P. Rajadurai, “A survey on working vacation queueing models,” Int. J. Pure Appl. Math, vol. 106, no. 6, pp. 33-41, 2016.
G. Ayyappan and K. Thilagavathy, “Analysis of MAP/PH/1 queueing model with breakdown, instantaneous feedback and server vacation,” Applications and Applied Mathematics: An International Journal (AAM), vol. 15, no. 2, p. 1, 2020.
A. Choudhary, S. R. Chakravarthy, and D. C. Sharma, “Analysis of MAP/PH/1 queueing system with degrading service rate and phase type vacation,” Mathematics, vol. 9, no. 19, p. 2387, 2021.
“A server backup model with markovian arrivals and phase type services,” European Journal of Operational Research, vol. 184, no. 2, pp. 584-609, 2008. doi: https://doi.org/10.1016/j.ejor.2006.12.016
D. Y. Yang and C. H. Wu, “Cost-minimization analysis of a working vacation queue with npolicy and server breakdowns,” Computers & Industrial Engineering, vol. 82, pp. 151-158, 2015.
K. Kalidass and R. Kasturi, “A queue with working breakdowns,” Computers & Industrial Engineering, vol. 63, no. 4, pp. 779-783, 2012.
Q. Ye and L. Liu, “Analysis of MAP/M/1 queue with working breakdowns,” Communications in Statistics-Theory and Methods, vol. 47, no. 13, pp. 3073-3084, 2018. [Online]. Available: https://doi.org/10.1080/03610926.2017.1346808
B. Deepa and K. Kalidass, “An M/M/1/N queue with working breakdowns and vacations,” International Journal of Pure and Applied Mathematics, vol. 119, no. 10, pp. 859-873, 2018.
R. Sharma, “Threshold n-policy for MX/H2/1 queueing system with un-reliable server and vacations,” J Int Acd Phy Sci, vol. 14, pp. 1-11, 2010.
C. J. Singh, M. Jain, and B. Kumar, “Analysis of M/G/1 queueing model with state dependent arrival and vacation,” Journal of Industrial Engineering International, vol. 8, pp. 1-8, 2012.
K.-H. Wang, K.-W. Chang, and B. Sivazlian, “Optimal control of a removable and nonreliable server in an infinite and a finite M/H2/1 queueing system,” Applied mathematical modelling, vol. 23, no. 8, pp. 651-666, 1999.
M. Jain, R. Sharma, and G. C. Sharma, “Multiple vacation policy for MX/Hk/1 queue with un-reliable server,” Journal of Industrial Engineering International, vol. 9, pp. 1-11, 2013.
G. Latouche and V. Ramaswami, Introduction to matrix analytic methods in stochastic modeling. SIAM, 1999.
S. Majid and P. Manoharan, “Analysis of an M/M/1 queue with working vacation and vacation interruption,” Applications and Applied Mathematics: An International Journal (AAM), vol. 14, no. 1, p. 2, 2019.
D. Gross, Fundamentals of queueing theory. John wiley & sons, 2008.
S. R. Chakravarthy and R. Kulshrestha, “A queueing model with server breakdowns, repairs, vacations, and backup server,” Operations research perspectives, vol. 7, pp. 100-131, 2020.
Downloads
Published
Issue
Section
License
Copyright (c) YUJOR
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
